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1
Sampling Distributions
2
Chapter Goals
After completing this chapter, you should be able to:
Define the concept of sampling error
Determine the mean and standard deviation for the sampling distribution of the sample mean, x
Determine the mean and standard deviation for the sampling distribution of the sample proportion, p
Describe the Central Limit Theorem and its importance
Apply sampling distributions for both x and p
_
_ _
_
3
Sampling Error
Sample Statistics are used to estimate Population Parameters
ex: X is an estimate of the
population mean, μ
Problems:
Different samples provide different estimates of the population parameter
Sample results have potential variability, thus sampling error exits
4
Calculating Sampling Error Sampling Error:
The difference between a value (a statistic) computed from a sample and the corresponding value (a parameter) computed from a population
Example: (for the mean)
where:
μ - xError Sampling
mean population μmean samplex
5
Review Population mean: Sample Mean:
N
xμ i
where:
μ = Population mean
x = sample mean
xi = Values in the population or sample
N = Population size
n = sample size
n
xx i
6
Example
If the population mean is μ = 98.6 degrees and a sample of n = 5 temperatures yields a sample mean of = 99.2 degrees, then the sampling error is
degrees0.699.298.6μx
x
7
Sampling Errors Different samples will yield different sampling
errors
The sampling error may be positive or negative ( may be greater than or less than μ)
The expected sampling error decreases as the
sample size increases
x
8
Sampling Distribution
A sampling distribution is a distribution of the possible values of a statistic for a given size sample selected from a population
9
Developing a Sampling Distribution
Assume there is a population …
Population size N=4
Random variable, x,
is age of individuals
Values of x: 18, 20,
22, 24 (years)
A B C D
10
.3
.2
.1
0 18 20 22 24
A B C DUniform Distribution
P(x)
x
(continued)
Summary Measures for the Population Distribution:
Developing a Sampling Distribution
214
24222018
N
xμ i
2.236N
μ)(xσ
2i
11
1st 2nd Observation Obs 18 20 22 24
18 18,18 18,20 18,22 18,24
20 20,18 20,20 20,22 20,24
22 22,18 22,20 22,22 22,24
24 24,18 24,20 24,22 24,24
16 possible samples (sampling with replacement)
Now consider all possible samples of size n=2
1st 2nd Observation Obs 18 20 22 24
18 18 19 20 21
20 19 20 21 22
22 20 21 22 23
24 21 22 23 24
(continued)
Developing a Sampling Distribution
16 Sample Means
12
1st 2nd Observation Obs 18 20 22 24
18 18 19 20 21
20 19 20 21 22
22 20 21 22 23
24 21 22 23 24
Sampling Distribution of All Sample Means
18 19 20 21 22 23 240
.1
.2
.3 P(x)
x
Sample Means
Distribution
16 Sample Means
_
Developing a Sampling Distribution
(continued)
(no longer uniform)
13
Summary Measures of this Sampling Distribution:
Developing aSampling Distribution
(continued)
2116
24211918
N
xμ i
x
1.5816
21)-(2421)-(1921)-(18
N
)μ(xσ
222
2xi
x
14
Comparing the Population with its Sampling Distribution
18 19 20 21 22 23 240
.1
.2
.3 P(x)
x 18 20 22 24
A B C D
0
.1
.2
.3
PopulationN = 4
P(x)
x_
1.58σ 21μxx2.236σ 21μ
Sample Means Distribution
n = 2
15
If the Population is Normal(THEOREM 6-1)
If a population is normal with mean μ and
standard deviation σ, the sampling distribution
of is also normally distributed with
and
x
μμx n
σσx
16
z-value for Sampling Distributionof x
Z-value for the sampling distribution of :
where: = sample mean= population mean= population standard deviation
n = sample size
xμσ
n
σμ)x(
z
x
17
Finite Population Correction Apply the Finite Population Correction if:
the sample is large relative to the population
(n is greater than 5% of N)
and… Sampling is without replacement
Then
1NnN
n
σ
μ)x(z
18
Normal Population Distribution
Normal Sampling Distribution (has the same mean)
Sampling Distribution Properties
(i.e. is unbiased )x x
x
μμx
μ
xμ
19
Sampling Distribution Properties
For sampling with replacement:
As n increases,
decreasesLarger sample size
Smaller sample size
x
(continued)
xσ
μ
20
If the Population is not Normal We can apply the Central Limit Theorem:
Even if the population is not normal, …sample means from the population will be
approximately normal as long as the sample size is large enough
…and the sampling distribution will have
and
μμx n
σσx
21
n↑
Central Limit Theorem
As the sample size gets large enough…
the sampling distribution becomes almost normal regardless of shape of population
x
22
Population Distribution
Sampling Distribution (becomes normal as n increases)
Central Tendency
Variation
(Sampling with replacement)
x
x
Larger sample size
Smaller sample size
If the Population is not Normal(continued)
Sampling distribution properties:
μμx
n
σσx
xμ
μ
23
How Large is Large Enough? For most distributions, n > 30 will give a
sampling distribution that is nearly normal
For fairly symmetric distributions, n > 15
For normal population distributions, the sampling distribution of the mean is always normally distributed
24
Example Suppose a population has mean μ = 8 and
standard deviation σ = 3. Suppose a random sample of size n = 36 is selected.
What is the probability that the sample mean is between 7.8 and 8.2?
25
ExampleSolution:
Even if the population is not normally distributed, the central limit theorem can be used (n > 30)
… so the sampling distribution of is approximately normal
… with mean = 8
…and standard deviation
(continued)
x
xμ
0.536
3
n
σσx
26
Example
Solution (continued):(continued)
x
0.31080.4)zP(-0.4
363
8-8.2
nσ
μ- μ
363
8-7.8P 8.2) μ P(7.8 x
x
z7.8 8.2 -0.4 0.4
Sampling Distribution
Standard Normal Distribution .1554
+.1554
x
Population Distribution
??
??
?????
??? Sample Standardize
8μ 8μx 0μz
27
Population Proportions, p p = the proportion of population having some characteristic
Sample proportion ( p ) provides an estimate of p:
If two outcomes, p has a binomial distribution
size sample
sampletheinsuccessesofnumber
n
xp
28
Sampling Distribution of p
Approximated by a
normal distribution if:
where
and
(where p = population proportion)
Sampling DistributionP( p )
.3
.2
.1 0
0 . 2 .4 .6 8 1 p
pμp
n
p)p(1σ
p
5p)n(1
5np
29
z-Value for Proportions If sampling is without replacement and n is greater than 5% of the
population size, then must use the finite population correction
factor:
1N
nN
n
p)p(1σ
p
np)p(1
pp
σ
ppz
p
Standardize p to a z value with the formula:
pσ
30
Example
If the true proportion of voters who support
Proposition A is p = .4, what is the
probability that a sample of size 200 yields a
sample proportion between .40 and .45?
i.e.: if p = .4 and n = 200, what is
P(.40 ≤ p ≤ .45) ?
31
Example if p = .4 and n = 200, what is
P(.40 ≤ p ≤ .45) ?
(continued)
.03464200
.4).4(1
n
p)p(1σ
p
1.44)zP(0
.03464
.40.45z
.03464
.40.40P.45)pP(.40
Find :
Convert to standard normal:
pσ
32
Example
z.45 1.44
.4251
Standardize
Sampling DistributionStandardized
Normal Distribution
if p = .4 and n = 200, what is
P(.40 ≤ p ≤ .45) ?
(continued)
Use standard normal table: P(0 ≤ z ≤ 1.44) = .4251
.40 0p